Question

valerie is going to purchase a new car. the car she wants has a list price of \\$32,495. valerie is planning to make a down payment of \\$1,877. furthermore, she plans to trade in her current car, which is a 2006 hyundai sonata in good condition. she will finance the rest of the cost by making monthly payments over five years. she can finance the cost at a rate of 8.64\\%, compounded monthly. she will also have to pay 8.23\\% sales tax, a \\$2,243 vehicle registration fee, and a \\$314 documentation fee. if the dealer gives valerie 87.5\\% of the trade-in price on her car, listed below, approximately how much will valerie pay in total for her new car? (round all dollar values to the nearest cent, and consider the trade-in to be a reduction in the amount paid.)

hyundai cars in good condition
\

$$\begin{tabular}{|l|c|c|c|c|} \\hline model/year & 2004 & 2005 & 2006 & 2007 \\\\ \\hline sonata & \\$6,145 & \\$6,520 & \\$6,784 & \\$7,066 \\\\ \\hline tiburon & \\$6,880 & \\$7,144 & \\$7,382 & \\$7,785 \\\\ \\hline elantra & \\$4,211 & \\$4,425 & \\$4,598 & \\$4,880 \\\\ \\hline accent & \\$5,576 & \\$5,828 & \\$6,005 & \\$6,317 \\\\ \\hline \\end{tabular}$$

a. \\$37,385
b. \\$38,821
c. \\$38,287
d. \\$36,944

Explanation:

Calculate trade-in value

Find the trade-in value of a 2006 Hyundai Sonata in good condition from the table:

$$ \text{Table Value} = \$6,784 $$

The dealer gives \(87.5\%\) of this value:

$$ \text{Trade-in Credit} = 6784 \times 0.875 = \$5,936.00 $$

Calculate total purchase cost

Calculate the total cost of the car including sales tax, registration, and documentation fees:

$$ \text{Sales Tax} = 32495 \times 0.0823 = \$2,674.3385 \approx \$2,674.34 $$

$$ \text{Total Cost} = 32495 + 2674.34 + 2243 + 314 = \$37,726.34 $$

Determine amount to finance

Subtract the down payment and the trade-in credit from the total cost:

$$ \text{Amount Financed } (P) = 37726.34 - 1877 - 5936 = \$29,913.34 $$

Calculate monthly loan payment

Use the monthly payment formula for an amortized loan:

$$ M = P \frac{r(1+r)^n}{(1+r)^n - 1} $$

Where:

• \(P = 29913.34\)
• Monthly interest rate \(r = \frac{0.0864}{12} = 0.0072\)
• Number of payments \(n = 5 \times 12 = 60\)

$$ M = 29913.34 \frac{0.0072(1.0072)^{60}}{(1.0072)^{60} - 1} \approx \$615.81 $$

Calculate total amount paid

Sum the down payment, the trade-in credit, and all monthly payments:

$$ \text{Total Payments} = 615.81 \times 60 = \$36,948.60 $$

$$ \text{Total Paid} = \text{Down Payment} + \text{Total Payments} = 1877 + 36948.60 = \$38,825.60 $$

Comparing with the options, this is closest to option (c) \(\$38,287\) or (b) \(\$38,821\). Let's re-verify if sales tax is calculated on the net price (after trade-in/down payment) or list price.
Usually, sales tax is calculated on the list price or list price minus trade-in.
If tax is on list price:

$$ \text{Tax} = 32495 \times 0.0823 = 2674.34 $$

If we recalculate with high precision:

$$ M = 29913.3385 \times \frac{0.0072 \times 1.0072^{60}}{1.0072^{60} - 1} = 29913.3385 \times 0.0205864 = 615.806 \approx 615.81 $$

$$ \text{Total Paid} = 1877 + (615.81 \times 60) = 38825.60 $$

If sales tax is calculated on the list price minus trade-in:

$$ \text{Taxable Amount} = 32495 - 5936 = 26559 $$

$$ \text{Tax} = 26559 \times 0.0823 = 2185.81 $$

$$ \text{Total Cost} = 32495 + 2185.81 + 2243 + 314 = 37237.81 $$

$$ \text{Amount Financed} = 37237.81 - 1877 - 5936 = 29424.81 $$

$$ M = 29424.81 \times 0.0205864 = 605.75 $$

$$ \text{Total Paid} = 1877 + (605.75 \times 60) = 38222 $$

If sales tax is calculated on list price, and trade-in is treated as a down payment reduction (not reducing tax):
Let's check option (b) \(\$38,821\):
If \(M = 615.73\), then \(615.73 \times 60 + 1877 = 38820.80 \approx 38821\).
This matches option (b) extremely closely.

Answer:

• (a) $37,385
• (b) $38,821 (Correct answer)
• (c) $38,287
• (d) $36,944